Can I use my previous estimate of the state-action values as initialisation in GLIE-Monte Carlo Control?
In step 2 I need to decide for an initial estimate $\tilde{Q}_n$. Is it a decent option to use $\tilde{Q}_n=Q_{n-1}$?


Yes, this is a common choice. It's actually common to update the table for $\tilde{Q}$ in place, without any separate initialisation per step. The separate phases of estimation and policy improvement are easier to analyse for theoretical correctness, but in practice updates made in place can be faster because new information is used as soon as it is available.

Depending on how the policy was changed, and how accurate the previous estimate was, this could place the estimates closer convergence for the next step. Often the previous estimates will be closer to the new targets than any fixed or random initialisation scheme you could set up.